Finance and Stochastics

, Volume 19, Issue 4, pp 763–790 | Cite as

Aggregation-robustness and model uncertainty of regulatory risk measures



Research related to aggregation, robustness and model uncertainty of regulatory risk measures, for instance, value-at-risk (VaR) and expected shortfall (ES), is of fundamental importance within quantitative risk management. In risk aggregation, marginal risks and their dependence structure are often modelled separately, leading to uncertainty arising at the level of a joint model. In this paper, we introduce a notion of qualitative robustness for risk measures, concerning the sensitivity of a risk measure to the uncertainty of dependence in risk aggregation. It turns out that coherent risk measures, such as ES, are more robust than VaR according to the new notion of robustness. We also give approximations and inequalities for aggregation and diversification of VaR under dependence uncertainty, and derive an asymptotic equivalence for worst-case VaR and ES under general conditions. We obtain that for a portfolio of a large number of risks, VaR generally has a larger uncertainty spread compared to ES. The results warn that unjustified diversification arguments for VaR used in risk management need to be taken with much care, and they potentially support the use of ES in risk aggregation. This in particular reflects on the discussions in the recent consultative documents by the Basel Committee on Banking Supervision.


Value-at-risk Expected shortfall Dependence uncertainty Risk aggregation Aggregation-robustness Inhomogeneous portfolio Basel III 

Mathematics Subject Classification

62G35 60E15 62P05 

JEL Classification




The authors would like to thank two referees, an Associate Editor and the Editor for helpful comments which have substantially improved the paper, and Edgars Jakobsons (ETH Zurich) for his kind help on some numerical examples in this paper. Paul Embrechts thanks the Oxford-Man Institute for its hospitality during his visit as 2014 Oxford-Man Chair. Ruodu Wang acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Forschungsinstitut für Mathematik (FIM) at ETH Zurich during his visits in 2013 and 2014.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsRiskLab and SFI, ETH ZurichZurichSwitzerland
  2. 2.Department of MathematicsBeijing Technology and Business UniversityBeijingChina
  3. 3.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada

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